In their 2009 simulation study comparing properties of Hodgkin Huxley vs. Markov Models (well worth a read) Martin and Denis discussed how an optimised short voltage step protocol might contain enough information to fit the parameters of models (termed an ‘identifiable’ model/protocol combination) in a relatively short amount of experimental time.

We picked up on these ideas when Kylie came to look at models of hERG. We originally wanted to study different modes of drug binding with hERG and design experiments to quantify that. Unfortunately, it rapidly became clear there was little consensus on how to model hERG itself, before even considering drug binding.

Figure 1: seven existing model structures that described the 29 models for IKr (a.k.a.* hERG) that we could find in the literature.

OK, so we have lots of different structures, but does this matter? Or do they all give similar predictions? Unfortunately – as we show in Figure 1 of the paper – quite a wide variety of different current profiles are predicted, even by models for the same species, cell type and temperature.

So Kylie’s PhD project became a challenge of deciding where we should start! What complexity do we need in model of hERG (for studying its role in the action potential and what happens when it is blocked), and how should we build one?

These questions link back to a couple of my previous posts – how complex should a model be, and what experiments do we need to do to build it? Kylie’s thesis looked at the question of how we should parameterise ion channel models, and even how to select the right ion channel model to use in the first place. We had quite a lot of fun designing new voltage clamp protocols and then going to a lab to test them out. The full story is in Kylie’s thesis, and we present a simpler version that just shows how well you can do with one basic model in the paper.

Kylie did a brilliant job, and as well as doing all the statistical inference and mathematical modelling work, she went and learnt how to do whole-cell patch clamp experiments herself as well at Teun de Boer’s lab and also with Adam Hill and Jamie Vandenberg. Patch clamp is an amazing experimental technique where you effectively get yourself an electrode in the middle of a cell, my sketch of how it works is in Figure 2.

Figure 2: the idea behind patch clamp. Top: you first make a glass pipette by pulling a glass tube whilst heating it, to melt it and stretch it until it breaks into two really fine pipettes (micrometers across at the end). You put one electrode in a bath with the cells, then you put another electrode in your pipette with some liquid; attach to a rig to get fine control of where it points; and lower it down under a microscope onto the surface of a single cell. You then apply suction, clamping the pipette to a cell, this is commonly done by literally sucking on a straw! Bottom: you then keep sucking, and rupture the cell membrane, this has cunningly got you an electric circuit that effectively lets you measure voltages or currents across the cell membrane. You can clamp a certain voltage at the amplifier, which it does by injecting current to keep a stable voltage (or any voltage as a function of time). The idea is that the current the amplifier has to inject is the exact opposite of what the cell itself is allowing across the membrane (give or take some compensation for the other electrical components I have put in my diagram). So you can now measure the current flowing through the cells ion channels as a function of voltage.

We decided that the traditional approach of specific fixed voltage steps (which neatly de-couples time- and voltage-dependence) was a bit slow and tricky to assemble into a coherent model. So we made up some new sinusoid-based protocols for the patch clamp amplifier to rapidly probe the voltage- and time-dynamics of the currents. Things we learnt along the way:

Whilst it might work in theory for the model, you also might fry the cells in real life (our first attempts at protocols went up to +100mV for extended periods of time, which cells don’t really like).

It’s really important to learn what all the dials on a patch clamp amplifier do(!), and adjust for things like liquid junction potential.

Synthetic data studies (simulating data, adding realistic levels of noise, and then attempting to recover the parameters you used) are a very useful tool for designing a good experiment. You can add in various errors and biases and see how sensitive your answers are to these discrepancies.

Despite conductance and kinetics being theoretically separable/identifiable, and practically in synthetic studies, we ended up with some problems here when using real data (e.g. kinetics make channel ‘twice as open’ with ‘half the conductance’. You can imagine this is impossible if the channels are already over 50% open, but maybe quite likely if only 5% of the channels are open?). We re-designed the voltage clamp to include an activation step to provoke a nice large current with a large open probability, based on hERG currents people had observed before.

But to cut a very long story short – it all worked better than we could have imagined. Figure 3 shows the voltage protocol we put into the amplifier, and the currents we recorded in CHO cells that were over-expressing hERG. We then fitted our simple Hodgkin-Huxley style model to the current, by varying all of its parameters to get the best possible fit, essentially.

Figure 3: Model Training/Fitting/Calibrating our model to currents provoked under the sinusoidal voltage clamp. Top: the whole training dataset. Bottom: a zoom in on the highlighted region of the top plot.

So a great fit, but that doesn’t mean anything on its own – see my previous post on that. So we then tested the model in situations that we would like it to make good predictions, here under cardiac action potentials and also slightly awry ones, see Fig 4.

Figure 4: Model Evaluation/Validation. The red current trace was recorded in the same cell as the sinusoidal data shown above. The blue trace is predicted using the parameters from the fitting exercise in Figure 3, and isn’t fitted to this recording.

We repeated this in a few different cells, and this lets us look at cell-cell variability in the ion channel kinetics via examining changes in the model’s parameters. Anyway, that is hopefully enough to whet your appetite for reading the whole paper! As usual, all the data, code, and (perhaps unusually) fitting algorithms are available for anyone to play with too.

Wish list: if you can help with any of these, let’s collaborate! Please get in touch.

A better understanding of identifiability of conductance versus kinetic parameters, and how to ensure both.

A way to design voltage clamp protocols for particular currents (this was somewhat hand tuned).

A way to select between different model structures at the same time as parameterising them.

A way to say how ‘similar’ (in terms of model dynamics?) a validation protocol is to a training protocol. If validation was too similar to training, it wouldn’t really be validation… we think our case above is ‘quite different’, but could we quantify this?

A way to quantify/learn ‘model discrepancy’ and to put realistic probabilistic bounds on our model predictions when we are using the models “out in the wild” in future unknown situations.

Footnotes

*hERG is the gene that encodes for the mRNA that is translated into a protein that assembles into homotetramers (groups of four of the same thing stuck together) in the cell membrane. This protein complex forms the main part of a channel in the cell membrane (Kv11.1) that carries the ionic current known as the “rapid delayed rectifier potassium current” or IKr. So you can see why we abuse the term hERG and say things like “hERG current”!